A spaceship is in space and at rest. To reach a speed of 2.90 × 10^3 m/s, what is the required velocity of the exhaust gas if the ship mass is 7.38 × 10^3 kg and the fuel is 400 kg?

To find the required velocity of the exhaust gas for the spaceship to reach a specific speed, we can use the principle of conservation of momentum as it applies to rockets, which is described by the Tsiolkovsky rocket equation. The equation shows the relationship between the velocity of the exhaust gas, the change in velocity of the rocket, and the mass of the rocket and its fuel.

In this scenario, the spaceship starts at rest and needs to achieve a final velocity of 2.90 × 10^3 m/s. The mass of the spaceship when fully loaded (including fuel) is 7.38 × 10^3 kg, and the mass of the fuel is 400 kg. The mass of the spaceship without fuel is therefore ( 7.38 \times 10^3 , \text{kg} - 400 , \text{kg} = 6.98 \times 10^3 , \text{kg} ).

Using the rocket equation, we can rearrange it to find the velocity of the exhaust gas:

[

v_e = \frac{(m_i - m_f) \cdot v}{m_f}

]

Where:

• ( v_e )